Question: Simplify the following expression and state the condition under which the simplification is valid. $k = \dfrac{p^3 - 4p}{-6p^2 - 24p - 24}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ k = \dfrac {p(p^2 - 4)} {-6(p^2 + 4p + 4)} $ $ k = -\dfrac{p}{6} \cdot \dfrac{p^2 - 4}{p^2 + 4p + 4} $ Next factor the numerator and denominator. $ k = - \dfrac{p}{6} \cdot \dfrac{(p + 2)(p - 2)}{(p + 2)(p + 2)}$ Assuming $p \neq -2$ , we can cancel the $p + 2$ $ k = - \dfrac{p}{6} \cdot \dfrac{p - 2}{p + 2}$ Therefore: $ k = \dfrac{ -p(p - 2)}{ 6(p + 2)}$, $p \neq -2$